p-pseudocompactness and related topics in topological spaces

We prove some basic properties of p-bounded subsets (p∈ω∗) in terms of z-ultrafilters and families of continuous functions. We analyze the relations between p-pseudocompactness with other pseudocompact like-properties as p-compactness and α-pseudocompactness where α is a cardinal number. We give an example of a sequentially compact ultrapseudocompact α-pseudocompact space which is not ultracompact, and we also give an example of an ultrapseudocompact totally countably compact α-pseudocompact space which is not q-compact for any q∈ω∗, answering affirmatively to a question posed by S. Garcı́a-Ferreira and Koc̆inac (1996). We show the distribution law clγ(X×Y)(A×B)=clγXA×clγYB, where γZ denotes the Dieudonné completion of Z, for p-bounded subsets and we generalize the classical Glisckberg Theorem on pseudocompactness in the realm of p-boundedness. These results are applied to study the degree of pseudocompactness in the product of p-bounded subsets.